On the Graph Homology with Integral Coefficients

In \cite{BS}, the authors constructa spectral sequence which converges to the homology groups of the graph configuration space. This construction requires a characteristic $0$ field to ensure a commutative model for the cochain algebra. For arbitrary coefficients a commutative model may not exist and we suggest a different approach. Each vertex of a graph $G$ is colored by a copy of $C_{N}^{*}(M;R)$, the normalized cochains, where $R$ is a commutative ring with unity of any characteristic. We construct a complex similar to the Bendersky-Gitler complex in \cite{BG}. Its differential involves sums over sequences of collapsing edges of the graph: for a single collapsed edge multiplication of tensor factors in $C_{N}^{*}(M;R)$ is used, while for general sequences one uses the sequence operations of McClure and Smith, cf. \cite{MS}. If the graph $G$ has at most 5 vertices with a planar-type labelling and $Z_{G} \subset M^{\times n}$ is the closed subspace of diagonals built from the graph $G$, we show this computes the relative cohomology groups $H^{*}(M^{\times n},Z_{G};R)$. These are isomorphic to the homology groups of the graph configuration space $H_{nm-*}(M^{G};R)$ if $M$ is a compact oriented manifold. When $G$ is the complete graph on $n$ vertices, these are the homology groups of the usual (labelled) configuration space